A connection on P is equivalent to a covariant derivative on H, which in turn can be ... The definition of a connection as a covariant derivative has an immediate ...
Linear Connections on Matrix Geometries
arXiv:hep-th/9411127v2 19 Dec 1994
J. Madore, T. Masson Laboratoire de Physique Th´eorique et Hautes Energies* Universit´e de Paris-Sud, Bˆ at. 211, F-91405 ORSAY
J. Mourad Laboratoire de Mod`eles de Physique Math´ematique Parc de Grandmont, Universit´e de Tours, F-37200 TOURS
Abstract: A general definition of a linear connection in noncommutative geometry has been recently proposed. Two examples are given of linear connections in noncommutative geometries which are based on matrix algebras. They both possess a unique metric connection.
LPTHE Orsay 94/96 November, 1994 * Laboratoire associ´e au CNRS. 1
1 Introduction and Motivation The extension to noncommutative algebras of the notion of a differential calculus has been given both without (Connes 1986) and with (Dubois-Violette 1988) use of the derivations of the algebra. A definition has been given (Chamseddine et al. 1993) of a possible noncommutative generalization of a linear connection which uses the left-module structure of the differential forms. Recently a different definition has been given (Mourad 1994, Dubois-Violette et al. 1994) which makes essential use of the full bimodule structure of the differential forms. We shall use this definition here to consider linear connections on two examples of noncommutative geometries based on matrix algebras. Both have a unique linear connection, which is metric and torsion free. In this respect they are similar to the quantum plane, which is not based on a finite-dimensional algebra. The general definition of a linear connection is given in this section and in Section 2 some basic formulae from matrix geometry are recalled. In Section 3 we consider an algebra of forms based on derivations and we show that there is a unique metric linear connections without torsion. This case is very similar to ordinary differential geometry and the calculations follow closely those of this section. In Section 4 we consider a more abstract differential geometry whose differential calculus is not based on derivations. Here we find that there is a unique 1-parameter family of connections, which is without torsion. The condition that the connection be metric fixes the value of the parameter. We first recall the definition of a linear connection in commutative geometry, in a form (Koszul 1960) which allows for a noncommutative generalization. Let V be a differential manifold and let (Ω∗ (V ), d) be the ordinary differential calculus on V . Let H be a vector bundle over V associated to some principle bundle P . Let C(V ) be the algebra of smooth functions on V and H the left C(V )-module of smooth sections of H. A connection on P is equivalent to a covariant derivative on H, which in turn can be characterized as a linear map D H → Ω1 (V ) ⊗C(V ) H (1.1) which satisfies the condition D(f ψ) = df ⊗ ψ + f Dψ
(1.2)
for arbitrary f ∈ C(V ) and ψ ∈ H. The definition of a connection as a covariant derivative has an immediate extension to noncommutative geometry. Let A be an arbitrary algebra and (Ω∗ (A), d) a differential calculus over A. We shall define in the next section a differential calculus (Ω∗ (Mn ), d) over the matrix algebras Mn . One defines a covariant derivative on a left A-module H as a map D
H → Ω1 (A) ⊗ H
(1.3)
which satisfies the condition (1.2) but with f ∈ A. A linear connection on V can be defined as a connection on the cotangent bundle to V . It can be characterized as a linear map D Ω1 (V ) → Ω1 (V ) ⊗C(V ) Ω1 (V ) (1.4) which satisfies the condition D(f ξ) = df ⊗ ξ + f Dξ
(1.5)
for arbitrary f ∈ C(V ) and ξ ∈ Ω1 (V ). Suppose, for simplicity that V is parallelizable and choose θα to be a globally defined moving frame on V . The connection form ω α β is defined in terms of the covariant derivative of the moving frame: Dθα = −ω α β ⊗ θβ .
(1.6)
Because of (1.5) the covariant derivative Dξ of an arbitrary element ξ = ξα θα ∈ Ω1 (V ) can be written as Dξ = (Dξα ) ⊗ θα where (1.7) Dξα = dξα − ω β α ξβ . Let π be the projection of Ω1 (V ) ⊗C(V ) Ω1 (V ) onto Ω2 (V ). The torsion form Θα can be defined as Θα = (d − πD)θα . 2
(1.8)
The derivative DX ξ along the vector field X, DX ξ = iX Dξ, 1
α
(1.9) α
β
is a linear map of Ω (V ) into itself. In particular DX θ = −ω β (X)θ . Using DX an extension of D can be constructed to the tensor product Ω1 (V ) ⊗C(V ) Ω1 (V ). We define DX (θα ⊗ θβ ) = DX θα ⊗ θβ + θα ⊗ DX θβ 1
(1.10)
1
Now let σ be the action on Ω (V ) ⊗C(V ) Ω (V ) defined by the permutation of two derivations: σ(ξ ⊗ η)(X, Y ) = ξ ⊗ η(Y, X)
(1.11)
and define σ12 = σ ⊗ 1. Then (1.10) can be rewritten without explicitly using the vector field as D(θα ⊗ θβ ) = Dθα ⊗ θβ + σ12 (θα ⊗ Dθβ ).
(1.12)
Define π12 = π ⊗ 1. If the torsion vanishes one finds that π12 D2 θα = −Ωα β ⊗ θβ where Ω
α
β
(1.13)
is the curvature 2-form. Notice that the equality π12 D2 (f θα ) = f π12 D2 θα
(1.14)
π(σ + 1) = 0.
(1.15)
is a consequence of the identity 1
The module Ω (V ) has a natural structure as a right C(V )-module and the corresponding condition equivalent to (1.5) is determined using the fact that C(V ) is a commutative algebra: D(ξf ) = D(f ξ).
(1.16)
D(ξf ) = σ(ξ ⊗ df ) + (Dξ)f.
(1.17)
Using σ this can also be written in the form By extension, a linear connection over a general noncommutative algebra A with a differential calculus (Ω∗ (A), d) can be defined as a linear map D
Ω1 (A) → Ω1 (A) ⊗A Ω1 (A)
(1.18)
1
which satisfies the condition (1.5) for arbitrary f ∈ A and ξ ∈ Ω (A). The module Ω1 (A) has again a natural structure as a right A-module but in the noncommutative case it is impossible in general to consistently impose the condition (1.16) and a substitute must be found. We consider first the case where the differential calculus (Ω∗ (A), d) is defined using the derivations of A (DuboisViolette 1988). Let X and Y be arbitrary derivations of A and suppose that the transposition σ in (1.11) maps Ω1 (A) ⊗A Ω1 (A) into itself. Then we propose to define D(ξf ) by the equation (1.17) (Dubois-Violette & Michor 1994a,b). A covariant derivative is a map of the form (1.18) which satisfies the Leibniz rules (1.5) and (1.17). The right Leibniz rule (1.18) can be made more transparent using the covariant derivative DX with respect to the derivation X. The two Leibniz rules can be written as DX (f ξ) = (Xf )ξ + f DX ξ, (1.19) DX (ξf ) = ξXf + (DX ξ)f. A metric g on V can be defined as a C(V )-bilinear, symmetric map of Ω1 (V ) ⊗C(V ) Ω1 (V ) into C(V ). This definition makes sense if one replaces C(V ) by an algebra A and Ω1 (V ) by a differential calculus Ω1 (A) over A. By analogy with the commutative case we shall say that the covariant derivative (1.17) is metric if the following diagram is commutative: Ω1 ⊗ A Ω1 g↓ A
D
−→ Ω1 ⊗A Ω1 ⊗A Ω1 ↓1⊗g d −→ Ω1
(1.20)
We have here set Ω1 (A) = Ω1 . In general symmetry must be defined with respect to the map σ. By a symmetric metric then we mean one which satisfies the condition gσ = g. 3
(1.21)
2 Matrix geometries Noncommutative geometry is based on the fact that one can formulate (Koszul 1960) much of the ordinary differential geometry of a manifold in terms of the algebra of smooth functions defined on it. It is possible to define a finite noncommutative geometry based on derivations by replacing this algebra by the algebra Mn of n× n complex matrices (Dubois-Violette et al. 1989, 1990). Since Mn is of finite dimension as a vector space, all calculations reduce to pure algebra. Matrix geometry is interesting in being similar is certain aspects to the ordinary geometry of compact Lie groups; it constitutes a transition to the more abstract formalism of general noncommutative geometry (Connes 1986, 1990). Our notation is that of Dubois-Violette et al. (1989). See also Madore (1994). In this section we recall some important formulae. Let λr , for 1 ≤ r ≤ n2 − 1, be an anti-hermitian basis of the Lie algebra of the special unitary group SUn in n dimensions. The λr generate Mn and the derivations er = ad λr form a basis for the Lie algebra of derivations Der(Mn ) of Mn . We define df for f ∈ Mn by df (er ) = er (f ).
(2.1)
(2.2)
In particular dλr (es ) = −C r st λt . We raise and lower indices with the Killing metric grs of SUn . We define the set of 1-forms Ω1 (Mn ) to be the set of all elements of the form f dg with f and g in Mn . The set of all differential forms is a differential algebra Ω∗ (Mn ). The couple (Ω∗ (Mn ), d) is a differential calculus over Mn . There is a convenient system of generators of Ω1 (Mn ) as a left- or right-module completely characterized by the equations θr (es ) = δsr . (2.3) The θr are related to the dλr by the equations dλr = C r st λs θt ,
θr = λs λr dλs .
(2.4)
The θr satisfy the same structure equations as the components of the Maurer-Cartan form on the special unitary group SUn : 1 dθr = − C r st θs θt . (2.5) 2 The product on the right-hand side of this formula is the product in Ω∗ (Mn ). We shall refer to the θr as a frame or Stehbein. If we define θ = −λr θr we can write the differential df of an element f ∈ Ω0 (Mn ) as a commutator: df = −[θ, f ]. (2.6)
4
3 A differential calculus with derivations From (2.5) we see that the linear connection defined by 1 ω r s = − C r st θt 2
Dθr = −ω r s ⊗ θs ,
(3.1)
has vanishing torsion. With this connection the geometry of Mn looks like the invariant geometry of the group SUn . It follows from the antisymmetry of C r st that Dθr = dθr . Since the elements of the algebra commute with the frame θr , we can define D on all of Ω∗ (Mn ) using (1.5). The map σ is given by σ(θr ⊗ θs ) = θs ⊗ θr . (3.2) It follows that D satisfies also (1.17). Consider a general covariant derivative. We can suppose it to be of the form Dθr = −ω r st θs ⊗ θt
(3.3)
with ω r st an arbitrary element of Mn for each value of (r, s, t). Then from (1.5) and (1.17) we find that 0 = D([f, θr ]) = [f, Dθr ]
(3.4)
and so the ω r st must be all in the center of Mn . They are complex numbers. If we define the torsion as in (1.8) and require that it vanish then we have ω r [st] = C r st .
(3.5)
Define a metric on Mn by the equation g(θr ⊗ θs ) = g rs . It satisfies the symmetry condition (1.21). The commutativity of the diagram (1.20) is the formal analogue of the condition that a connection be metric. If we impose it we see that ω r (st) = 0. (3.6) The linear connection (3.1) is the unique torsion-free metric connection on Ω1 (Mn ). From the formula analogous to (1.13) we find that the curvature 2-form is given by Ωr s =
1 r C st C t uv θu θv . 8
The connection (3.1) has been used (Dubois-Violette et al. 1989, Madore 1990, Madore & Mourad 1993, Madore 1994) in the construction of noncommutative generalizations of Kaluza-Klein theories. In particular the Dirac operator has a natural coupling to it, determined by a correspondence principle.
5
4 A differential calculus without derivations Equation (1.17) can be extended in principle to the case of a differential calculus which is not based on derivations if we postulate (Mourad 1994) the existence of a map σ
Ω1 ⊗A Ω1 −→ Ω1 ⊗A Ω1
(4.1)
to replace the one defined by (1.11). We define then D(ξf ) by the equation (1.17) but using (4.1) instead of (1.11). From the identity D (ξf )g = D ξ(f g) we find that σ be right A-linear. From the identity D d (f g)h = D d f (gh) we find that σ be also left A-linear (Dubois-Violette et al. 1994). In general σ 2 6= 1.
(4.2)
The extension of D to Ω1 ⊗ Ω1 is given by (1.12) but with σ defined by (4.1). As an example we shall consider a differential calculus over an algebra of matrices with a differential defined by a graded commutator (Connes & Lott 1990). Consider the matrix algebra Mn with a ZZ2 grading. One can define on Mn a graded derivation dˆ by the formula ˆ = −[θ, f ], df
(4.3)
where θ is an arbitrary anti-hermitian odd element and the commutator is taken as a graded commutator. ˆ = −2θ2 and for any α ∈ Mn , We find that dθ dˆ2 α = [θ2 , α].
(4.4)
The ZZ2 grading of Mn can be expressed as the direct sum Mn = Mn+ ⊕ Mn− where Mn+ (Mn− ) are the even (odd) elements of Mn . It can be induced from a decomposition Cn = Cl ⊕ Cn−l for some integer l. The elements of Mn+ are diagonal with respect to the decomposition; the elements of Mn− are off-diagonal. It is possible to construct over Mn+ a differential algebra Ω∗ = Ω∗ (Mn+ ) (Connes & Lott 1991). Let ˆ Ω0 = Mn+ and let Ω1 ≡ dΩ0 ⊂ Mn− be the Mn+ -bimodule generated by the image of Ω0 in Mn− under d. Define d (4.5) Ω0 −→ Ω1 ˆ Let dΩ1 be the M + -module generated by the image of Ω1 in M + under d. ˆ It using directly (4.3): d = d. n n 2 1 would be natural to try to set Ω = dΩ and define d
Ω1 −→ Ω2
(4.6)
ˆ 1 . If we using once again (4.3). Every element of Ω1 can be written as a sum of elements of the form f0 df attempt to define an application (4.6) using again directly (4.3), ˆ 1 + f0 dˆ2 f1 , ˆ 1 ) = df ˆ 0 df d(f0 df
(4.7)
then we see that in general d2 does not vanish. To remedy this problem we eliminate simply the unwanted terms. Let Im dˆ2 be the submodule of dΩ1 consisting of those elements which contain a factor which is the image of dˆ2 and define Ω2 by Ω2 = dΩ1 /Im dˆ2 . (4.8) 6
Then by construction the second term on the right-hand side of (4.7) vanishes as an element of Ω2 and we have a well defined map (4.6) with d2 = 0. This procedure can be continued to arbitrary order by iteration. For each p ≥ 2 we let Im dˆ2 be the submodule of dΩp−1 defined as above and we define Ωp by Ωp = dΩp−1 /Im dˆ2 .
(4.9)
Since Ωp Ωq ⊂ Ωp+q the complex Ω∗ is a differential algebra. The Ωp need not vanish for large values of p. In fact if θ2 ∝ 1 we see that dˆ2 = 0 and the sequence defined by (4.9) never stops. However Ωp ⊆ Mn+ (Mn− ) for p even (odd) and so it stabilizes for large p. We shall consider in some detail the case n = 3 with the grading defined by the decomposition C3 = 2 C ⊕ C. The most general possible form for θ is θ = η1 − η1∗ where
0 0 η1 = 0 0 0 0
(4.10)
a b. 0
(4.11)
Without loss of generality we can choose the euclidean 2-vector η1i of unit length. The general construction yields Ω0 = M3+ = M2 × M1 and Ω1 = M3− but after that the quotient by elements of the form Im dˆ2 reduces the dimension. One finds Ω2 = M1 and Ωp = 0 for p ≥ 3. Let e be the unit of M1 . It generates Ω2 and can also be considered as an element of Ω0 . To form a basis for Ω1 we must introduce a second matrix η2 . It is convenient to choose it of the same form as η1 . We have then in Ω2 the identity ηi ηj∗ = 0. We shall further impose that ηi∗ ηj = δij e.
(4.12)
It follows that dη1 = e,
dη2 = 0.
We can uniquely fix η2 by requiring that there be a unitary element u ∈ M2 ⊂ M3+ which exchanges η1 and η2 : η2 = uη1 , η1 = −uη2 . (4.13) We have also η2 u = 0,
η1 u = 0.
(4.14)
The vector space of 1-forms is of dimension 4 over the complex numbers. The dimension of Ω ⊗C Ω1 is equal to 16 but the dimension of the tensor product Ω1 ⊗M + Ω1 is equal to 5. One finds in fact over M3+ 3 the relations ηi ⊗ ηj = 0, ηi∗ ⊗ ηj∗ = 0, (4.15) η2∗ ⊗ η1 = 0, η1∗ ⊗ η2 = 0, η2∗ ⊗ η2 = η1∗ ⊗ η1 . 1
which leave ηij = ηi ⊗ ηj∗ ,
ζ = η1∗ ⊗ η1
(4.16)
as independent basis elements. We can make therefore the identification Ω1 ⊗M + Ω1 = M3+ = Ω0 . 3
(4.17)
To define a covariant derivative we must first introduce the map σ of (4.1). Because of the left and right M3+ -linearity the map σ is entirely determined by its action on ζ and, for example, η11 : X aij ηij + aζ, σ(η11 ) = ij
σ(ζ) =
X
bij ηij + bζ.
ij
7
If we multiply both sides of the second equation by u we find that bij = 0; if we multiply both sides of the first equation by u2 we find that a = 0. Let v be a matrix such that vη1 = η1 and vη2 6= η2 . From the conditions of left and right linearity we have the equations σ(η11 ) = vσ(η11 ) =
X
σ(η11 ) = σ(η11 )v ∗ =
aij vηij ,
X
aij ηij v ∗ ,
ij
ij
from which we conclude that a11 = µ,
a12 = a21 = a22 = 0,
where µ is an arbitrary complex number. If we impose the condition (1.15) we find that 1 + b = 0. So σ is given by σ(η11 ) = µη11 , σ(ζ) = −ζ. (4.18) The Hecke relation (σ + 1)(σ − µ) = 0 is satisfied. Suppose that µ 6= −1 and define Λ∗ (S ∗ ) to be the quotient of the tensor algebra by the ideal generated by the eigenvectors of µ (−1). As a complex vector space Λ∗ is of dimension 10. The map ζ 7→ e induces an isomorphism of Λ∗ with Ω∗ . As a complex vector space S ∗ is of dimension 13. It is an unusual fact that it is of finite dimension. If µ = −1 then σ = −1 also and (1.15) is trivially satisfied. In this case it is natural to define Λ∗ to be the entire tensor algebra. On the universal differential calculus the projection π of (1.15) is the identity and σ must be equal to −1. The covariant derivative of ηi must be of the form Dηi =
X
cijk ηjk + ci ζ.
jk
The exterior derivative of u is given by du = η2 − η2∗ .
(4.19)
From (1.5) we find then that D must satisfy the constraints Dη1 = ζ − uDη2 ,
Dη2 = uDη1
(4.20)
and therefore that c1 = 1, c2 = 0 and c2ij is determined in terms of c1ij : c211 = −c121 ,
c212 = −c122 ,
c221 = c111 ,
c222 = c112 .
From the condition (1.17) one finds the additional constraints (Dη1 )u − σ(η12 ) = 0.
(Dη2 )u − σ(η22 ) = 0,
which both imply that c111 = −µ,
c112 = 0,
c121 = 0.
c122 = 0
(4.21)
The covariant derivative is uniquely defined then in terms of σ: Dη1 = −µη11 + ζ,
Dη2 = −µη21 .
(4.22)
The lack of symmetry is due to the fact that the form θ which determines the exterior derivative is defined in terms of η1 . The torsion vanishes. Recently (Dubois-Violette et al. 1994) the quantum plane has been shown to possess a 1-parameter family of covariant derivatives, which also are torsion free. If one takes the covariant derivative of the identity θe = η1 and its adjoint one finds that Dη1∗ = −η11 − ζ, 8
Dη2∗ = −η12 ,
(4.23)
and therefore that Dθ = (σ − 1) θ ⊗ θ.
(4.24)
One finds also that D(de) = (1 + µ)η11 ,
D(du) = η12 − µη21 .
(4.25)
Using the identification (4.17) one sees that ˆ 1 = −θ ⊗ η1 − η1 ⊗ θ = η11 + ζ, dη ˆ ∗ = −θ ⊗ η ∗ − η ∗ ⊗ θ = −η11 − ζ dη 1
1
(4.26)
1
ˆ Therefore when µ = −1 one can identify D with d. Let g be a metric and set hij = g(ηij ),
h = g(ζ).
(4.27)
If we suppose that the metric is bilinear then hij is given in terms of h11 . For example h21 = uh11 .
(4.28)
The condition that the connection be metric compatible is expressed by the equations dh11 = −µη1 h + η1∗ h11 , dh = −η1 h + µη1∗ h11 .
(4.29)
This equation has no solutions unless µ2 = 1. If µ = 1 then to within an overall scale the unique bilinear metric is given by hij = ηi ηj∗ , h = −e, (4.30) where the right-hand sides are considered as elements of M3+ . Therefore hij (h) takes its values in the M2 (M1 ) factor of M3+ . From (4.18) we see that the metric (4.30) is not symmetric. With the normalization we have chosen the frames ηi have unit norm with respect to the metric: Tr(hij ) = δij .
(4.31)
The curvature can be defined by a formula analogous to (1.13). Using (1.12) we find Dη11 = ζ ⊗ η1∗ − µη11 ⊗ η1 , Dη12 = ζ ⊗ η2∗ , Dη21 = −µη21 ⊗ η1 , Dζ = µζ ⊗ η1∗ − η11 ⊗ η1 ,
(4.32)
from which we conclude that D2 η1 = (µ2 − 1)η11 ⊗ η1 , D2 η1∗ = (µ + 1)(η11 ⊗ η1 − ζ ⊗ η1∗ ),
D2 η2 = µ2 η21 ⊗ η1 , D2 η2∗ = −ζ ⊗ η2∗ .
(4.33)
The curvature is given by the projection of D2 η1 onto Ω2 ⊗M + Ω1 : 3
π12 D2 η1 = 0, π12 D2 η2 = 0, 2 ∗ ∗ π12 D η1 = −(µ + 1)e ⊗ η1 , π12 D2 η2∗ = −e ⊗ η2∗ .
(4.34)
Although by construction the operator π12 D2 is left linear it is not right linear. For no value of µ does the curvature vanish. However the analogue of the square of the curvature tensor does vanishes. In fact, the tensor product of the curvature tensor with itself vanishes identically. There are 4 different frames corresponding to the 4 different ways of choosing ηi and ηi∗ as generators of Ω1 . The action of the matrix u which takes one into the other is a change of frame. Since hij is not proportional 9
to the identity matrix, the frames cannot be considered as the analogues of orthonormal frames and since h22 6= h11 the change of frame u cannot be considered as ‘orthonormal’. If we define ¯ (i) e ⊗ η ∗ , π12 D2 ηi∗ = −R i
π12 D2 ηi = −R(i) e ⊗ ηi ,
(4.35)
we see that R vanishes in all frames and that ¯ (1) = µ + 1, R
¯ (2) = 1. R
(4.36)
¯ from µ + 1 to 1. When we take η1∗ into η2∗ by the right action of u we change the value of R ∗ Since η1 = θe and η1 = −eθ we could also choose θ as frame. If we rewrite Equation (4.35) is this frame, π12 D2 θ = −R(θ) e ⊗ θ,
(4.37)
we see that the component of the curvature can be defined by one matrix, proportional to the identity matrix, given by R(θ) = µ + 1. (4.38) The analogue of the Ricci tensor would be obtained by using the metric to ‘contract two indices’ of the curvature tensor. We can do this here if we identify Ω2 with the vector space Λ2 . We can define a left-linear map Ric of Ω1 into itself by Ric(ξ) = −(1 ⊗ g)π12 D2 ξ. (4.39) From, for example, the identity (1 ⊗ g)(ζ ⊗ η1∗ ) = η1∗ h11 = η1∗
(4.40)
ones sees that Ric is given by the equations Ric(ηi ) = R(i) ηi ,
¯ (i) ηi∗ . Ric(ηi∗ ) = R
(4.41)
The geometry is therefore not ‘Ricci-flat’. There is no analogue of the Ricci scalar. There does not seem to be any way to construct a frame-independent quantity so the best we can do is declare θ to be a preferred frame and consider the component (4.38) of the curvature in this frame as the curvature of the geometry of M3+ . If we require that it be metric the connection is unique and any action would yield it as extremal. We could on the other hand consider µ as an unknown parameter and chose as action 2 ) = 3(µ + 1)2 . (4.42) Tr(R(θ) The action has then a minimal which corresponds to a connection which is not metric and whose curvature component vanishes in the frame θ. Additional structure could be put on the algebra M3+ . For example one could replace the M2 component with the algebra of quaternions or require that the matrices be hermitian. In the latter case the two possible frames are η1 + η1∗ and η2 + η2∗ . They yield each one curvature component whose values are given by (4.36). We mentioned that the Dirac operator has a natural coupling to the geometry of the previous section. There is also a generalized correspondence principle which can be used as a guide in introducing the Dirac operator coupled to the geometry of the quantum plane. The coupling of the Dirac operator to the geometry considered here is however more problematic. There is no possible correspondence principle since the geometry is not a deformation of a commutative geometry. It is natural to require that a spinor be an element of a left M3 -module and that the Dirac operator be an hermitian element of M3− but otherwise there is no restriction. In ordinary geometry the exterior derivative can be identified with the commutator of the Dirac operator (Connes 1986) and this has been used as motivation for proposing iθ as the Dirac operator in the present case, without any consideration of curvature (Connes & Lott 1990). This or any other element of M3− could be considered as automatically coupled to the curvature since there is a unique metric connection. Since we have a differential calculus we have an associated cohomology. By definition H 0 = M3+ and H p = 0 for p ≥ 3. The unique 2-cocyle e is the coboundary of η1 and so H 2 = 0. The vector space Z 1 of 1-cocycles is of complex dimension 2, generated by η2 and η2∗ . From (4.19) we see that η2 − η2∗ is a coboundary; it is easy to verify that so also is η2 + η2∗ . Therefore H 1 = 0 and the cohomology is trivial. Acknowledgment: The authors would like to thank M. Dubois-Violette for enlightening comments. 10
References Chamseddine A.H., Felder G., Fr¨ohlich J. 1993, Gravity in Non-Commutative Geometry, Commun. Math. Phys. 155 205. Connes A. 1986, Non-Commutative Differential Geometry, Publications of the Inst. des Hautes Etudes Scientifique. 62 257. — 1990, G´eom´etrie noncommutative, InterEditions, Paris. Connes A., Lott J. 1990, Particle Models and Noncommutative Geometry, in ‘Recent Advances in Field Theory’, Nucl. Phys. Proc. Suppl. B18 29. — 1991, The metric aspect of non-commutative geometry, Proceedings of the Carg`ese Summer School, (to appear). Dubois-Violette M. 1988, D´erivations et calcul diff´erentiel non-commutatif, C. R. Acad. Sci. Paris 307 S´erie I 403. Dubois-Violette M., Kerner R., Madore J. 1989, Gauge bosons in a noncommutative geometry, Phys. Lett. B217 485; Classical bosons in a noncommutative geometry, Class. Quant. Grav. 6 1709. — 1990, Noncommutative differential geometry of matrix algebras, J. Math. Phys. 31 316. Dubois-Violette M., Madore J., Masson T., Mourad J. 1994, Linear Connections on the Quantum Plane, Preprint LPTHE Orsay 94/94. Dubois-Violette M., Michor P. 1994a, D´erivations et calcul diff´erentiel non-commutatif II, C. R. Acad. Sci. Paris 319 S´erie I 927. Dubois-Violette M., Michor P. 1994b, Private communication. Koszul J.L. 1960, Lectures on Fibre Bundles and Differential Geometry, Tata Institute of Fundamental Research, Bombay. Madore J. 1990, Modification of Kaluza-Klein Theory, Phys. Rev. D41 3709. Madore J. 1994, An Introduction to Noncommutative Differential Geometry and its Physical Applications, Cambridge University Press (to appear). Madore J., Mourad J. 1993, Algebraic-Kaluza-Klein Cosmology, Class. Quant. Grav. 10 2157. Mourad. J. 1994, Linear Connections in Non-Commutative Geometry, Univ. of Tour Preprint.
11
